“This is so rare to get these opportunities,” coach Pete Carroll says of the 6-foot-7 receiver and 2013 All-Pro who has 171 catches and 26 touchdowns the last two seasons. Seattle used four tight ends last season. They barely caught half as many balls as Graham’s 85 receptions.

Saturday was the first day NFL teams could begin negotiating with free agents. Seattle general manager John Schneider was phoning counterparts with other teams, as is his normal routine, to find out what might be available and who may be shedding what for salary cap purposes.

The New Orleans Saints told him they were interested in exploring a potential trade of Jimmy Graham, their three-time Pro Bowl and 2013 All-Pro tight end.

Hello!

The Seahawks had just terminated the contract of starting tight end Zach Miller the day before because of a failed physical following two ankle surgeries. They were pursuing Denver’s Julius Thomas, but Jacksonville wanted him more and spent more to woo him there. They were perhaps interested in Cleveland free agent Jordan Cameron, but he’s been sidelined by multiple concussions.

Graham? The matchup nightmare with a basketball-like ability to snare balls out of the air, 171 of them for 26 touchdowns during the last two seasons? Suddenly Seattle’s tight-end shopping shifted from Macy’s to Saks Fifth Avenue.

“This was one of the players that was brought up,” Schneider said. “And we continued to pursue it.”

All the way through the biggest splash of the brand-new league year.

New Orleans agreed to send its superstar tight end plus a fourth-round draft choice to Seattle for Seahawks’ two-time Pro Bowl center Max Unger and their first-round pick next month. The deal came together minutes before the league year began at 1 p.m. Pacific Time on Tuesday.

“This is a fantastic job by John to get this lined up. This is so rare to get these opportunities, to get that done,” Carroll said.

“Yeah, it’s a continued message we are competing. You’ve got to keep moving forward. And it takes big decisions like these to do that.”

The Seahawks also announced the signing of free agent cornerback Cary Williams from Philadelphia to replace Byron Maxwell, who signed for $70 million, $25 million guaranteed with the Eagles. Williams gets $18 million for three years in Seattle.

Schneider said the Seahawks valued the 30-year-old Williams, who won the Super Bowl while with Baltimore three seasons ago, for his length and height (6 feet 1) in playing press coverage.

Graham’s arrival gives Seattle one of the most dynamic tight ends around. The Seahawks used four tight ends last season because of injuries.

The 28-year-old Graham, a six-year veteran who is 6 feet 7 and 260 pounds, was on Schneider’s and Seattle’s radar for the 2010 draft before going to New Orleans in the third round that year. He has 85 and 86 catches with 10 and 16 touchdowns in each of the last two seasons. Seattle’s four tight ends last season had 48 catches and six touchdowns combined.

“We’ve added a guy who is a big-time difference maker at his position,” Schneider said.

Seattle ranked 20th in red-zone scoring of touchdowns last season, at a rate of 51.7 percent.

“I think he has 46 touchdowns since 2011,” Carroll said, accurately. “He’s a big target. He has the basketball background. He has the ability to get off the ground and play high. He’s got a real sense for finishing plays.

“Just a fantastic talent and a great weapon for us. I’m sure he’ll be a great benefit for our passing game and for the productivity that we want down there. We always want to run the football, but we need those targets. And he brings us an obvious opportunity to get the ball in the end zone.”

The league’s highest-paid tight end signed a four-year, $40 million deal with New Orleans before last season — after he sued the Saints to try to be classified as a wide receiver for purposes of a potential franchise tag that never came to be. He was most often in the slot as an inside wide receiver, while not playing a traditional tight end for nearly two-thirds of his snaps in New Orleans recently.

But Carroll said Graham will be a tight end in Seattle, a dynamic one who has improved his blocking recently and is still working on it.

He is scheduled to earn $2.9 million in a guaranteed base salary with a $5 million roster bonus due for the Seahawks to pay Thursday. His base salary for 2016 is a nonguaranteed $8.9 million. For 2017 it’s $7.9 million.

As for the homework the Seahawks did before making this deal as compared to the team’s last huge offseason trade, in 2013 for Percy Harvin from Minnesota that also sent away Seattle’s first-round draft pick, Schneider said: “We think we learned from our previous deal with Percy.”

Graham comes at a steep cost befitting his value.

Unger was the foundation of an at-times shaky Seahawks offensive line, its chief communicator who had a strong bond with Wilson. Unger has two years remaining on the contract he signed before the 2012 season. The Seahawks drafted him in the second round in 2009 out of Oregon. His base salary for this year is $4.5 million.

After adding Graham’s $8 million cap hit and subtracting Unger‘s, the deal appears to add $4.6 million to the Seahawks’ 2015 salary cap and leaves them with an estimated $14.4 million to spend this year. That $4.6 million is nearly the amount the Seahawks saved by terminating Miller’s contract Friday. Miller’s cap cost was scheduled to be $3.39 million this year, and Seattle absorbed a $1 million acceleration cost to cut him.

Essentially the Seahawks released Miller, plus traded Unger and their 31st-overall draft choice for Graham and New Orleans’ fourth-round choice for a total added cap cost of $2.21 million this year.

With the league’s salary cap beginning at $143 million for 2015, that’s almost a wash.

And a coup.

“In some respects it actually helps us” financially, Schneider said. “It turns out a first-round draft choice costs close to three and a half million. And to go out and sign a tight end to a big bonus … would cost more.

“We are on course. We are on budget. We have a lot of goals, a lot to accomplish (yet this offseason).”

Unger’s departure is a signal to how much the Seahawks line coach Tom Cable likes Patrick Lewis at center. Unger’s recent backup of recent years, Lemuel Jeanpierre, is a restricted free agent. Lewis took Jeanpierre’s job and started four games last season when Unger missed 10 games with foot and knee injuries. He drew Cable’s praise for strength and adaptability.

As great as Unger was communicating along the line, being the unit’s glue and working in synchronization with Wilson, he missed 13 games the last two seasons because of injury — yet the Seahawks still were first and fourth in the NFL in rushing those last two seasons largely without him.

“We are going to figure it out as we go,” Carroll said of the center position. “We hope to let the competition take over. We’ve got depth.

“But we are going to miss Max.”

Perhaps not as much as they intend to feature Graham, though.

EXTRA POINTS

As expected, unrestricted free agent James Carpenter departed, signing with the New York Jets for a reported four years at about $4.75 million per season. So Seattle is in the market for a new left guard. Miami released Shelley Smith, and the veteran with 11 career starts in four seasons, is visiting the Seahawks Wednesday according to ESPN and NBC’s Pro Football Talk. Schneider said Alvin Bailey would be the left guard if Seattle “had to play a game tomorrow” — then noted how early it is in the shopping season. Free agency runs through April 24. The draft begins April 30. … The Jets released WR Percy Harvin, as expected. So Seattle gets New York’s sixth-round pick next month. ... The Oakland Raiders opened free agency by agreeing to contracts with Seattle linebacker and former Super Bowl MVP Malcolm Smith.